Comptes Rendus
no. 9-10
Mathematical Economics
Option price with stochastic volatility for both fast and slow mean-reverting regimes
[Prix de lʼoption à volatilité stochastique pour les régimes de retour à la moyenne, rapide ou lent]

Présenté par : Philippe G. Ciarlet

Qiang Zhang1 ; Jiguang Han2, 3, 1 ; Ming Gao1
1 Department of Mathematics, City University of Hong Kong, Hong Kong
2 USTC–CityU Joint Advanced Research Center, Suzhou, China
3 Department of Statistics and Finance, University of Science and Technology of China, China
Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 411-414.

Le modèle de volatilité stochastique de Heston a été largement utilisé dans la théorie financière moderne, en particulier pour déterminer le prix des options. Habituellement, ce modèle peut prendre en compte deux régimes différents : le régime de retour rapide à la moyenne et celui de retour lent à la moyenne. Deux solutions différentes ont été données, selon le régime du modèle. Nous démontrons un résultat surprenant : les deux solutions peuvent être approchées par une formule identique. Dans chaque régime, les prédictions de lʼapproximation sont très proches des solutions numériques du modèle de Heston.

The Heston model of stochastic volatility has been widely adopted in modern finance, especially in option pricing. Usually, the model can be classified as being in one of two different regimes: the fast mean-reverting regime and the slow mean-reverting regime. Different approximations are needed for each regime. We show a surprising result: the solution in both regimes can be approximated by an identical expression. The predictions of the approximation are in excellent agreement with the numerical solutions of the Heston model in both regimes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.05.008
Qiang Zhang 1 ; Jiguang Han 2, 3, 1 ; Ming Gao 1

1 Department of Mathematics, City University of Hong Kong, Hong Kong
2 USTC–CityU Joint Advanced Research Center, Suzhou, China
3 Department of Statistics and Finance, University of Science and Technology of China, China 
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Qiang Zhang; Jiguang Han; Ming Gao. Option price with stochastic volatility for both fast and slow mean-reverting regimes. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 411-414. doi : 10.1016/j.crma.2013.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.05.008/

[1] C.M. Bender; S.A. Orzag Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag, New York, 1999

[2] Jean Pierre Fouque; Georage Papanicolaou; K. Ronnie Sircar; Knut Solna Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, New York, 2011

[3] Jiguang Han; Ming Gao; Qiang Zhang; Yutian Li Option prices under stochastic volatility, Appl. Math. Lett., Volume 26 (2013) no. 1, pp. 1-4

[4] Steven L. Heston A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., Volume 6 (1993) no. 2, pp. 327-343

[5] Roger W. Lee Option pricing by transform methods: Extensions, unification, and error control, J. Comput. Finance, Volume 7 (2004) no. 3

[6] Roger Lord; Christian Kahl Optimal Fourier inversion in semi-analytical option pricing, J. Comput. Finance, Volume 10 (2007) no. 4

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